I have a question regarding convex polytopes. Let us say I have the vertices of a polytope which I name as $ V = \{v_1,\cdots,v_k\}$. Each of the $v_k$ are n-dimensional vectors, i.e. $v_k \in R^n$. I would like to know if it is possible to write the polytope as intersection of half-spaces using the information from the vertices, i.e. can I write the polytops as $Ax\leq b, A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m, x \in R^n$, where $m$ denotes the number of linear inequalities. Columns of A are not necessarily the vertices of the given polytope. An example, consider a polytope in $R^2_+$ with vertices $\{(0,1),(1,1),(2,0),(0,0)\}$. It can be observed that the corresponding half space representation is $Ax\le b$, where $$A=\begin{pmatrix}\\\
0 & 1 \\\ 1 & 1 \\\ -1 & 0\\\ 0 & -1\end{pmatrix}, b = (1,2,0,0 )^T$$.

Ziegler has more than one book, but I'm sure Dan means the one "Lectures on polytopes", which does discuss going back and forth between $V$-representation and $H$-representation (i.e. vertex representation and hyperplane representation) of a polytope.
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Patricia HershNov 14 '12 at 23:27

I will try to get the book and see the relevant sections.
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user27396Nov 14 '12 at 23:53

2 Answers
2

The problem you identify is called the facet enumeration problem in the literature: Given the vertices, find a description of the facets.
There has been quite a bit of work on this. For $n$ points in $d$ dimensions,
$O(n^{\lfloor d/2 \rfloor})$ is achievable, and aymptotically worstcase optimal.
But this is a theoretical result. The work of Avis & Fukuda, to which Igor refers,
is quite practical, achieving a complexity of $O(d^{O(1)} n M)$ where $M$ is the size
of the output description. Here is one reference: